Many spiritual teachers say that there are three bodies. The physical body, the astral body, and the causal body. The physical body of course is just the actual matter that makes up our body. The astral body is what enlivens the body, brings life and energy to the body. The causal body would be like the soul (hard to describe what that means).

Don't know if this is what was behind the meaning of your dream, but thought I would throw it out as one possibility.

my spiritual teacher has often said something to the effect of : we are three-in-one. The person we think we are, the person others think we are and the ONE we really are. I tried to find the exact wording but cannot find it at the moment.
Shanti!

Three has to do with commitment to the subject matter of the dream. It stands for mind, body and soul and is a request to commit mind body and soul to improving yourself in the direction indicated in the dream. Three will often appear in dreams about the heart as people who have suppressed their feelings often lack passion and commitment for new things in life. It is also similar to a broken 8 in appearance and 8 always symbolizes the heart. A triangle or pyramid also indicate commitment and request the dreamer to meditate on the subject matter of the dream.

The number 3 associates with God – the Holy Trinity. Three adult people can simply thus represent a good or godly person as your mind sees them to be. Three babies would represent a good or godly new idea, pet project, or new concept about someone, again as your mind sees it to be.

As your 3 adults or 3 babies are one person divided into three, this strongly associates with the nature of God. Yet, if the people you saw were all identical, however, the good or godly person – or the good idea they can represent – is really only good or godly according to your own ego’s concept of what is good and godly. This is so because while God is a Trinity – 3 persons in one – each aspect of the Lord is differentiated. There is God the Father, God the Holy Ghost and the Son, Jesus.

Note how Nature itself shows us this through the symbol of water which is usually used to represent The Spirit or God. We have 3 forms of water - liquid water - God the Father, gas/steam – the Holy Spirit, the power aspect of God, and a solid(ice) – the manifested God in the physical Son, Jesus.